Search Results (55)

In this integrative review paper, we explore the parallels between the physical models of gyrotrons and some equations used in diverse and broad scientific fields. These include Adler’s famous equation, Van der Pol equation, the Lotka–Volterra equations and the Kuramoto model. The paper is written in the form of a pedagogical discourse and aims to provide additional insights into gyrotron physics through analogies and parallels to theoretical approaches used in other fields of research. For the first time, reachability analysis is used in the context of gyrotron physics as a modern tool for understanding the behavior of nonlinear dynamical systems. Full article

For systems such as the van der Pol and van der Pol–Duffing oscillators, the study of their oscillation is currently a very active area of research. Many authors have used the bifurcation method to try to determine oscillatory behavior. But when the system involves n separate delays, the equations for bifurcation become quite complex and difficult to deal with. In this paper, the existence of periodic oscillatory behavior was studied for a system consisting of n coupled equations with multiple delays. The method begins by rewriting the second-order system of differential equations as a larger first-order system. Then, the nonlinear system of first-order equations is linearized by disregarding higher-degree terms that are locally small. The instability of the trivial solution to the linearized equations implies the instability of the nonlinear equations. Periodic behavior often occurs when the system is unstable and bounded, so this paper also studied the boundedness here. It follows from previous work on the subject that the conditions here did result in periodic oscillatory behavior, and this is illustrated in the graphs of computer simulations. Full article

We present recent advances in the analysis of nonlinear problems involving singular (degenerate) operators. The results are obtained within the framework of p-regularity theory, which has been successfully developed over the past four decades. We illustrate the theory with applications to degenerate problems in various areas of mathematics, including optimization and differential equations. In particular, we address the problem of describing the tangent cone to the solution set of nonlinear equations in singular cases. The structure of p-factor operators is used to propose optimality conditions and to construct novel numerical methods for solving degenerate nonlinear equations and optimization problems. The numerical methods presented in this paper represent the first approaches targeting solutions to degenerate problems such as the Van der Pol differential equation, boundary-value problems with small parameters, and partial differential equations where Poincaré’s method of small parameters fails. Additionally, these methods may be extended to nonlinear degenerate dynamical systems and other related problems. Full article

The 3-dimensional modified van der Pol–Duffing oscillator has been studied by several authors. We complete its study, first characterizing its zero-Hopf equilibria and then its zero-Hopf bifurcations—i.e., we provide sufficient conditions for the existence of three, two or one periodic solutions, bifurcating from the zero-Hopf equilibrium localized at the origin of coordinates. Recall that an equilibrium point of a 3-dimensional differential system whose eigenvalues are zero and a pair of purely imaginary eigenvalues is a zero-Hopf equilibrium. Finally, we determine the dynamics of this system near infinity, i.e., we control the orbits that escape to or come from the infinity. Full article

To deal with uncertainties in a dynamic system, many nonlinear control approaches have been considered. Unique challenges arise from uncertainties that are bounded by constants, which has led to the development of both continuous and discontinuous control methods. However, these methods either are limited to classes of smooth nonlinear models or have a tendency to result in chattering during practical applications. In this work, a novel auxiliary robust integral of the sign of the error (ARISE) controller is proposed to prevent chattering and deal with uncertainties (even those bounded by constants) for general, switched, and nonsmooth control-affine nonlinear systems. The ARISE control system includes a unique auxiliary error that is designed to inject a sliding mode (SM) term directly into the error system without including an SM term in the controller itself. In fact, the ARISE control law includes an integral SM term that is continuous. Consequently, the ARISE control law minimizes the chattering effect that results from discontinuous SM terms. The proposed ARISE control system is augmented with an adaptive update law to deal with the unknown control effectiveness matrix in the dynamic model. To prove the effectiveness of the ARISE controller, a nonlinear stability analysis was conducted and resulted in semi-global exponential tracking towards an ultimate bound. Furthermore, the performance of the proposed controller was evaluated and compared against a traditional SM controller through simulations using a switched Van der Pol oscillator model. It was concluded that the proposed ARISE controller performs better for a switched system than an SM controller. The improved performance of the ARISE controller was consistent across different dynamic parameters and disturbances. Full article

In this study, we examine the bifurcations and dynamics of a piecewise linear van der Pol equation—a model that captures self-sustained oscillations and is applied in various scientific disciplines, including electronics, neuroscience, biology, and economics. The van der Pol equation is transformed into a piecewise linear system to simplify the analysis of stability and controllability, which is particularly beneficial in engineering applications. This work explores the impact of increasing the number of linear segments on the system’s dynamics, focusing on the stability of the equilibria, phase portraits, and bifurcations. The findings reveal that while the bifurcation structure at critical values of the bifurcation parameter is complex, the topology of the piecewise linear model remains unaffected by an increase in the number of linear segments from three to four. This research contributes to our understanding of the dynamics of nonlinear systems with piecewise linear characteristics and has implications for the analysis and design of real-world systems exhibiting such behavior. Full article

The conventional winding number method is extended in this research to characterize the nonlinear dynamical systems, especially in differentiating partially predictable chaos from strong chaos. On modern robotics’ challenges with increased degrees of freedom, traditional methods like the Lyapunov exponent are insufficient for distinguishing between strong and partially predictable chaos. The proposed methods examine the winding number’s sensitivity with respect to the center and its standard deviations across time sequences to assess predictability and differentiate between different motion types. The Duffing–Van der Pol system is used to show the effectiveness in identifying different chaotic behaviours, offering significant implications for the control of complex robotic systems. Full article

This article focuses on the study of local truncation errors (LTEs) in the Network Simulation Method (NSM), specifically when using the trapezoidal method and Gear’s methods. The NSM, which represents differential equations through electrical circuit elements, offers advantages in solving nonlinear dynamic systems such as the van der Pol equation. The analysis compares the performance of these numerical methods in terms of their stability and error minimization, with particular emphasis on LTE. By leveraging circuit-based techniques prior to numerical application, the NSM improves convergence. This study evaluates the impact of step size on LTE and highlights the trade-offs between accuracy and computational cost when using the trapezoidal and Gear methods. Full article

This paper explores secure communication systems with a chaotic carrier. The use of chaotic oscillations instead of regular van der Pol oscillators as a signal carrier is a promising and active research area, providing not only communication systems with new protection principles and organization but also high steganographic efficiency when transmitting short messages. The problem is to select methods and techniques for mixing a useful signal into a chaotic one and its recovery on the receiver side, featuring a set of properties acceptable for implementation and real-world application. We demonstrate application of synergetic control theory (SCT), which provides advanced observer-basing methods for nonlinear dynamic systems as well as explore example of data transmission system consisting of a Genesio–Tesi chaotic oscillator, data signal transmission with a method of nonlinear modulation, and recovering with a single-channel synergetic observer at the receiver side. The paper presents a nonlinear state observer modeling procedure followed by building a MATLAB/Simulink simulation model of the data transmission system for the PC-platform along with software implementation for the Raspberry Pi platform, with simulation and experimental run results showing data transmission rates seem to be acceptable for the considered practical applications. Practical applications and limitations issues are discussed. Future research will be universal modeling procedures for different classes of chaotic generators and whole system experimental hardware implementation. The obtained results can be primarily used in short messages and/or encryption keys secure transmission systems, cyber-physical system component command communications, as well as chaotic carrier system R&D competitive studies and other applications. Full article

Motivated by important applications to the analysis of complex noise-induced phenomena, we consider a problem of the constructive description of randomly forced equilibria for nonlinear systems with multiplicative noise. Using the apparatus of the first approximation systems, we construct an approximation of mean square deviations that explicitly takes into account the presence of multiplicative noises, depending on the current system state. A spectral criterion of existence and exponential stability of the stationary second moments for the solution of the first approximation system is presented. For mean square deviation, we derive an expansion in powers of the small parameter of noise intensity. Based on this theory, we derive a new, more accurate approximation of mean square deviations in a general nonlinear system with multiplicative noises. This approximation is compared with the widely used approximation based on the stochastic sensitivity technique. The general mathematical results are illustrated with examples of the model of climate dynamics and the van der Pol oscillator with hard excitement. Full article

In this study, we investigate Quantum Long Short-Term Memory and Quantum Gated Recurrent Unit integrated with Variational Quantum Circuits in modeling complex dynamical systems, including the Van der Pol oscillator, coupled oscillators, and the Lorenz system. We implement these advanced quantum machine learning techniques and compare their performance with traditional Long Short-Term Memory and Gated Recurrent Unit models. The results of our study reveal that the quantum-based models deliver superior precision and more stable loss metrics throughout 100 epochs for both the Van der Pol oscillator and coupled harmonic oscillators, and 20 epochs for the Lorenz system. The Quantum Gated Recurrent Unit outperforms competing models, showcasing notable performance metrics. For the Van der Pol oscillator, it reports MAE 0.0902 and RMSE 0.1031 for variable x and MAE 0.1500 and RMSE 0.1943 for y; for coupled oscillators, Oscillator 1 shows MAE 0.2411 and RMSE 0.2701 and Oscillator 2 MAE is 0.0482 and RMSE 0.0602; and for the Lorenz system, the results are MAE 0.4864 and RMSE 0.4971 for x, MAE 0.4723 and RMSE 0.4846 for y, and MAE 0.4555 and RMSE 0.4745 for z. These outcomes mark a significant advancement in the field of quantum machine learning. Full article

The motivation behind this study is to overcome the complex mathematical formulation and time-consuming nature of traditional numerical methods used in solving differential equations. It seeks an alternative approach for more efficient and simplified solutions. A Deep Neural Network (DNN) is utilized to understand the intricate correlations between the oscillator’s variables and to precisely capture their dynamics by being trained on a dataset of known oscillator behaviors. In this work, we discuss the main challenge of predicting the behavior of oscillators without depending on complex strategies or time-consuming simulations. The present work proposes a favorable modified form of neural structure to improve the strategy for simulating linear and nonlinear harmonic oscillators from mechanical systems by formulating an ANN as a DNN via an appropriate oscillating activation function. The proposed methodology provides the solutions of linear and nonlinear differential equations (DEs) in differentiable form and is a more accurate approximation as compared to the traditional numerical method. The Van der Pol equation with parametric damping and the Mathieu equation are adopted as illustrations. Experimental analysis shows that our proposed scheme outperforms other numerical methods in terms of accuracy and computational cost. We provide a comparative analysis of the outcomes obtained through our proposed approach and those derived from the LSODA algorithm, utilizing numerical techniques, Adams–Bashforth, and the Backward Differentiation Formula (BDF). The results of this research provide insightful information for engineering applications, facilitating improvements in energy efficiency, and scientific innovation. Full article

In the paper, we first develop a novel automatically energy-preserving scheme (AEPS) for the undamped and unforced single and multi-coupled Duffing equations by recasting them to the Lie-type systems of ordinary differential equations. The AEPS can automatically preserve the energy to be a constant value in a long-term free vibration behavior. The analytical solution of a special Duffing–van der Pol equation is compared with that computed by the novel group-preserving scheme (GPS) which has fourth-order accuracy. The main novelty is that we constructed the quadratic forms of the energy equations, the Lie-algebras and Lie-groups for the multi-coupled Duffing oscillator system. Then, we extend the GPS to the damped and forced Duffing equations. The corresponding algorithms are developed, which are effective to depict the long term nonlinear vibration behaviors of the multi-coupled Duffing oscillators with an accuracy of $\mathcal{O}\left(h^4\right)$ for a small time stepsize h. Full article

In this paper, we applied a chaos control method based on integro-differential equations for stabilization of an unstable cardiac rhythm, which is described by a variation of the modified Van der Pol equation. Chaos control with this method may be useful for stabilization of irregular heartbeat using a small perturbation. This method differs from other stabilization strategies by the absence of adjustable parameters and the lack of rough approximations in determining control functions whose control parameters are fixed by the properties of the unstable system itself. Full article

In this paper, a system of four coupled van der Pol oscillators with delay is studied. Firstly, the conditions for the existence of multiple periodic solutions of the system are given. Secondly, the multiple periodic solutions of spatiotemporal patterns of the system are obtained by using symmetric Hopf bifurcation theory. The normal form of the system on the central manifold and the bifurcation direction of the bifurcating periodic solutions are derived. Finally, numerical simulations are attached to demonstrate our theoretical results. Full article